Recently, I set up the integral again, with care taken over the normalization of 4-dimensional vectors, and I concluded the approximation of gravity is actually inverse square! In other words: In the Half Earth universe, gravitational force is inverse cube in 4 dimensions, but because an object receives gravitational force from the past and future, the classical approximation of gravitational force is actually inverse square. So I need to rewrite some parts of Half Earth astronomy.
The Half Earth universe is a 4+1-dimensional universe with 4 dimensions of spacetime and 1 dimension of metatime. In our universe's spacetime, we change between inertial frames of reference using the Lorentz transformation, where space and time is transformed by a hyperbolic rotation (using sin and, cosh), and the Lorentz transformation preserves causality. In the Half Earth universe, frame of reference transformation is done by doing a conventional rotation (using sin and cos), we call it angular transformation.
Finally, don't forget about the Galilean transformation, where frame of reference transformations only shear the position component while leading time intact.
- Simultaneity: same
- Distance: same
- Time interval: same
- Interval: ds^2 = dx^2 + 0*dt^2
- Causality: preserved
- Constant-acceleration trajectory: parabola
- Rotation: Shear
- Matrix: [[1 v] [0 1]] = exp(v*[[0 1] [0 0]])
- Angle: v = v
- Trig: opposite=v, adjacent=1
- Algebra: dual numbers (i^2 = 0)
- Simultaneity: leading clock lags
- Distance: shorter
- Time interval: shorter
- Interval: ds^2 = dx^2 - dt^2
- Causality: preserved
- Constant-acceleration trajectory: hyperbola
- Rotation: Hyperbolic
- Matrix: [[cosh(beta) sinh(beta)] [sinh(beta) cosh(beta)]] = exp(beta*[[0 1] [1 0]])
- Angle: tanh beta = v
- Trig: opposite=sinh beta, adjacent=cosh beta
- Algebra: split-complex numbers (i^2 = 1)
- Simultaneity: leading clock leads
- Distance: longer
- Time interval: longer
- Interval: ds^2 = dx^2 + dt^2
- Causality: not preserved
- Constant-acceleration trajectory: elliptical arc
- Rotation: Circular
- Matrix: [[cos(theta) sin(theta)] [-sin(theta) cos(theta)]] exp(theta*[[0 1] [-1 0]])
- Angle: tan theta = v
- Trig: opposite=sin theta, adjacent=cos theta
- Algebra: complex numbers (i^2 = -1)
A universe with angular relativity will have relativistic effects opposite to our universe's Lorentz relativity. In our universe, time pass slower for moving objects, and in Half Earth universe, time pass faster for moving objects. In our universe, leading clocks lag, and in Half Earth universe, leading clocks lead. Finally, in Half Earth universe, moving objects are longer, rather than shorter.
The most striking differences between Lorentz transformations and angular transformation are (1) speed of light and (2) causality. In our universe, the light cone divide our hyperbolic space into spacelike and timelike regions, and the light cone imposes the cosmic speed limit and prevents sequences of events from reversing themselves.
In angular relativity, there is no cosmic speed limit: You can get faster and faster forever, but as you accelerate, something strange happens: your speed will approach infinity relative to rest frame, and then you can travel back in time! In other words, you are moving "backward" in time relative to a stationary observer: that is time travel. Since you are time traveling, the station observer can see TWO copies of you, one traveling forward in time and one backward in time. You can even go in circles and meet you in the past, of course. The conversion factor between distance and time unit is called c, and coincidentally, it is the launch speed of some small subatomic particles including light (many particles are launches 45 degrees away form source in space-time diagrams).
Time traveling leads to paradoxes, but there is a solution: The universe is self-adjusting over the 5th dimension called metatime. As metatime passes, the paths of the particles will adjust themselves so they approach the paths of least action. If we look at conservative forces where force = -grad potential, then a physical system's metatime evolution can be described as derivative (position over metatime) = -constant * ( force + grad potential ). The metatime differential equation follows the principle of relativity: It is true no matter what angular transformation (rotating in four dimensions, and picking an origin) you perform on the physical system.
The self-adjusting process may not be totally stable: That means, the universe is constantly adjusting paths of particles, but it can never reach a stage where no further adjustments can be made, even with the absence of time travelers.